Estimating the Distance between Two LocationsEstimating
the Area of ColombiaUsing
Map Scales and the Pythagorean Theorem

General George S. PattonUnited State Military AcademyWest Point, New York

This exercise using a map and a cursor asks the viewer
to find the distance
between two points.

For example, a student might be asked to find a distance
from Bogota
to the Panama Canal. First the viewer would click on Bogota
and note the coordinates at the upper right. These numbers
represent
pixels with an origin (0,0) at the lower left.

This is followed by clicking on the region most likely
to be the Panama
Canal and noting its coordinates. The viewer then uses the
distance
formula, to find the measure in pixels from Bogota to the Canal.

Finally, using these measurements and the map scale, the
viewer converts
the distance to miles or kilometers. The viewer uses the scale to
determine how to convert pixel distances on the map to ground measures
in miles or kilometers.

The same JAVA Lite Applet map/graphic permits viewers to
discover for
themselves several different ways to estimate the area of an irregular
region.

While both of these exercises are relatively simple
mathematics, estimating
is good for students of all ages. Moreover, finding areas of irregular
regions is background for integral calculus. In addition, the
distance
formula is derived from the Pythagorean Theorem.

The U.S. Central Intelligence Agency (CIA) provided the
map for this
exercise. Should you choose to adapt this script, we suggest that
you see the following link:

Dava Sobel, Longitude:
The true story
of a lone genius who solved the greatest scientific problem of his
time.
Fourth Estate, London, 1995.

Distances on a plane surface are simple. Finding accurate
distances,
especially in sailing east and west, have been difficult in past
centuries.
Much of this more complicated mathematics took place at Greenwich
Observatory
on the south bank of the Thames near London.

The NCB thanks Dr. Frank
WattenbergDepartment of Mathematical
Sciences at the
United States Military Academy for Deposit #8.